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      SUBROUTINE <a name="CLAGS2.1"></a><a href="clags2.f.html#CLAGS2.1">CLAGS2</a>( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV,
     $                   SNV, CSQ, SNQ )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      LOGICAL            UPPER
      REAL               A1, A3, B1, B3, CSQ, CSU, CSV
      COMPLEX            A2, B2, SNQ, SNU, SNV
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="CLAGS2.17"></a><a href="clags2.f.html#CLAGS2.1">CLAGS2</a> computes 2-by-2 unitary matrices U, V and Q, such
</span><span class="comment">*</span><span class="comment">  that if ( UPPER ) then
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">            U'*A*Q = U'*( A1 A2 )*Q = ( x  0  )
</span><span class="comment">*</span><span class="comment">                        ( 0  A3 )     ( x  x  )
</span><span class="comment">*</span><span class="comment">  and
</span><span class="comment">*</span><span class="comment">            V'*B*Q = V'*( B1 B2 )*Q = ( x  0  )
</span><span class="comment">*</span><span class="comment">                        ( 0  B3 )     ( x  x  )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  or if ( .NOT.UPPER ) then
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">            U'*A*Q = U'*( A1 0  )*Q = ( x  x  )
</span><span class="comment">*</span><span class="comment">                        ( A2 A3 )     ( 0  x  )
</span><span class="comment">*</span><span class="comment">  and
</span><span class="comment">*</span><span class="comment">            V'*B*Q = V'*( B1 0  )*Q = ( x  x  )
</span><span class="comment">*</span><span class="comment">                        ( B2 B3 )     ( 0  x  )
</span><span class="comment">*</span><span class="comment">  where
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">    U = (     CSU      SNU ), V = (     CSV     SNV ),
</span><span class="comment">*</span><span class="comment">        ( -CONJG(SNU)  CSU )      ( -CONJG(SNV) CSV )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">    Q = (     CSQ      SNQ )
</span><span class="comment">*</span><span class="comment">        ( -CONJG(SNQ)  CSQ )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Z' denotes the conjugate transpose of Z.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The rows of the transformed A and B are parallel. Moreover, if the
</span><span class="comment">*</span><span class="comment">  input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
</span><span class="comment">*</span><span class="comment">  of A is not zero. If the input matrices A and B are both not zero,
</span><span class="comment">*</span><span class="comment">  then the transformed (2,2) element of B is not zero, except when the
</span><span class="comment">*</span><span class="comment">  first rows of input A and B are parallel and the second rows are
</span><span class="comment">*</span><span class="comment">  zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  UPPER   (input) LOGICAL
</span><span class="comment">*</span><span class="comment">          = .TRUE.: the input matrices A and B are upper triangular.
</span><span class="comment">*</span><span class="comment">          = .FALSE.: the input matrices A and B are lower triangular.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A1      (input) REAL
</span><span class="comment">*</span><span class="comment">  A2      (input) COMPLEX
</span><span class="comment">*</span><span class="comment">  A3      (input) REAL
</span><span class="comment">*</span><span class="comment">          On entry, A1, A2 and A3 are elements of the input 2-by-2
</span><span class="comment">*</span><span class="comment">          upper (lower) triangular matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B1      (input) REAL
</span><span class="comment">*</span><span class="comment">  B2      (input) COMPLEX
</span><span class="comment">*</span><span class="comment">  B3      (input) REAL
</span><span class="comment">*</span><span class="comment">          On entry, B1, B2 and B3 are elements of the input 2-by-2
</span><span class="comment">*</span><span class="comment">          upper (lower) triangular matrix B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CSU     (output) REAL
</span><span class="comment">*</span><span class="comment">  SNU     (output) COMPLEX
</span><span class="comment">*</span><span class="comment">          The desired unitary matrix U.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CSV     (output) REAL
</span><span class="comment">*</span><span class="comment">  SNV     (output) COMPLEX
</span><span class="comment">*</span><span class="comment">          The desired unitary matrix V.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  CSQ     (output) REAL
</span><span class="comment">*</span><span class="comment">  SNQ     (output) COMPLEX
</span><span class="comment">*</span><span class="comment">          The desired unitary matrix Q.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      REAL               A, AUA11, AUA12, AUA21, AUA22, AVB11, AVB12,
     $                   AVB21, AVB22, CSL, CSR, D, FB, FC, S1, S2, SNL,
     $                   SNR, UA11R, UA22R, VB11R, VB22R
      COMPLEX            B, C, D1, R, T, UA11, UA12, UA21, UA22, VB11,
     $                   VB12, VB21, VB22
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           <a name="CLARTG.95"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>, <a name="SLASV2.95"></a><a href="slasv2.f.html#SLASV2.1">SLASV2</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, AIMAG, CMPLX, CONJG, REAL
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Statement Functions ..
</span>      REAL               ABS1
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Statement Function definitions ..
</span>      ABS1( T ) = ABS( REAL( T ) ) + ABS( AIMAG( T ) )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span>      IF( UPPER ) THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Input matrices A and B are upper triangular matrices
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Form matrix C = A*adj(B) = ( a b )
</span><span class="comment">*</span><span class="comment">                                   ( 0 d )
</span><span class="comment">*</span><span class="comment">
</span>         A = A1*B3
         D = A3*B1
         B = A2*B1 - A1*B2
         FB = ABS( B )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Transform complex 2-by-2 matrix C to real matrix by unitary
</span><span class="comment">*</span><span class="comment">        diagonal matrix diag(1,D1).
</span><span class="comment">*</span><span class="comment">
</span>         D1 = ONE
         IF( FB.NE.ZERO )
     $      D1 = B / FB
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        The SVD of real 2 by 2 triangular C
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">         ( CSL -SNL )*( A B )*(  CSR  SNR ) = ( R 0 )
</span><span class="comment">*</span><span class="comment">         ( SNL  CSL ) ( 0 D ) ( -SNR  CSR )   ( 0 T )
</span><span class="comment">*</span><span class="comment">
</span>         CALL <a name="SLASV2.132"></a><a href="slasv2.f.html#SLASV2.1">SLASV2</a>( A, FB, D, S1, S2, SNR, CSR, SNL, CSL )
<span class="comment">*</span><span class="comment">
</span>         IF( ABS( CSL ).GE.ABS( SNL ) .OR. ABS( CSR ).GE.ABS( SNR ) )
     $        THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute the (1,1) and (1,2) elements of U'*A and V'*B,
</span><span class="comment">*</span><span class="comment">           and (1,2) element of |U|'*|A| and |V|'*|B|.
</span><span class="comment">*</span><span class="comment">
</span>            UA11R = CSL*A1
            UA12 = CSL*A2 + D1*SNL*A3
<span class="comment">*</span><span class="comment">
</span>            VB11R = CSR*B1
            VB12 = CSR*B2 + D1*SNR*B3
<span class="comment">*</span><span class="comment">
</span>            AUA12 = ABS( CSL )*ABS1( A2 ) + ABS( SNL )*ABS( A3 )
            AVB12 = ABS( CSR )*ABS1( B2 ) + ABS( SNR )*ABS( B3 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           zero (1,2) elements of U'*A and V'*B
</span><span class="comment">*</span><span class="comment">
</span>            IF( ( ABS( UA11R )+ABS1( UA12 ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.152"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CMPLX( VB11R ), CONJG( VB12 ), CSQ, SNQ,
     $                      R )
            ELSE IF( ( ABS( VB11R )+ABS1( VB12 ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.155"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CMPLX( UA11R ), CONJG( UA12 ), CSQ, SNQ,
     $                      R )
            ELSE IF( AUA12 / ( ABS( UA11R )+ABS1( UA12 ) ).LE.AVB12 /
     $               ( ABS( VB11R )+ABS1( VB12 ) ) ) THEN
               CALL <a name="CLARTG.159"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CMPLX( UA11R ), CONJG( UA12 ), CSQ, SNQ,
     $                      R )
            ELSE
               CALL <a name="CLARTG.162"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CMPLX( VB11R ), CONJG( VB12 ), CSQ, SNQ,
     $                      R )
            END IF
<span class="comment">*</span><span class="comment">
</span>            CSU = CSL
            SNU = -D1*SNL
            CSV = CSR
            SNV = -D1*SNR
<span class="comment">*</span><span class="comment">
</span>         ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute the (2,1) and (2,2) elements of U'*A and V'*B,
</span><span class="comment">*</span><span class="comment">           and (2,2) element of |U|'*|A| and |V|'*|B|.
</span><span class="comment">*</span><span class="comment">
</span>            UA21 = -CONJG( D1 )*SNL*A1
            UA22 = -CONJG( D1 )*SNL*A2 + CSL*A3
<span class="comment">*</span><span class="comment">
</span>            VB21 = -CONJG( D1 )*SNR*B1
            VB22 = -CONJG( D1 )*SNR*B2 + CSR*B3
<span class="comment">*</span><span class="comment">
</span>            AUA22 = ABS( SNL )*ABS1( A2 ) + ABS( CSL )*ABS( A3 )
            AVB22 = ABS( SNR )*ABS1( B2 ) + ABS( CSR )*ABS( B3 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           zero (2,2) elements of U'*A and V'*B, and then swap.
</span><span class="comment">*</span><span class="comment">
</span>            IF( ( ABS1( UA21 )+ABS1( UA22 ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.188"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CONJG( VB21 ), CONJG( VB22 ), CSQ, SNQ, R )
            ELSE IF( ( ABS1( VB21 )+ABS( VB22 ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.190"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CONJG( UA21 ), CONJG( UA22 ), CSQ, SNQ, R )
            ELSE IF( AUA22 / ( ABS1( UA21 )+ABS1( UA22 ) ).LE.AVB22 /
     $               ( ABS1( VB21 )+ABS1( VB22 ) ) ) THEN
               CALL <a name="CLARTG.193"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CONJG( UA21 ), CONJG( UA22 ), CSQ, SNQ, R )
            ELSE
               CALL <a name="CLARTG.195"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( -CONJG( VB21 ), CONJG( VB22 ), CSQ, SNQ, R )
            END IF
<span class="comment">*</span><span class="comment">
</span>            CSU = SNL
            SNU = D1*CSL
            CSV = SNR
            SNV = D1*CSR
<span class="comment">*</span><span class="comment">
</span>         END IF
<span class="comment">*</span><span class="comment">
</span>      ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Input matrices A and B are lower triangular matrices
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Form matrix C = A*adj(B) = ( a 0 )
</span><span class="comment">*</span><span class="comment">                                   ( c d )
</span><span class="comment">*</span><span class="comment">
</span>         A = A1*B3
         D = A3*B1
         C = A2*B3 - A3*B2
         FC = ABS( C )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        Transform complex 2-by-2 matrix C to real matrix by unitary
</span><span class="comment">*</span><span class="comment">        diagonal matrix diag(d1,1).
</span><span class="comment">*</span><span class="comment">
</span>         D1 = ONE
         IF( FC.NE.ZERO )
     $      D1 = C / FC
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">        The SVD of real 2 by 2 triangular C
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">         ( CSL -SNL )*( A 0 )*(  CSR  SNR ) = ( R 0 )
</span><span class="comment">*</span><span class="comment">         ( SNL  CSL ) ( C D ) ( -SNR  CSR )   ( 0 T )
</span><span class="comment">*</span><span class="comment">
</span>         CALL <a name="SLASV2.229"></a><a href="slasv2.f.html#SLASV2.1">SLASV2</a>( A, FC, D, S1, S2, SNR, CSR, SNL, CSL )
<span class="comment">*</span><span class="comment">
</span>         IF( ABS( CSR ).GE.ABS( SNR ) .OR. ABS( CSL ).GE.ABS( SNL ) )
     $        THEN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute the (2,1) and (2,2) elements of U'*A and V'*B,
</span><span class="comment">*</span><span class="comment">           and (2,1) element of |U|'*|A| and |V|'*|B|.
</span><span class="comment">*</span><span class="comment">
</span>            UA21 = -D1*SNR*A1 + CSR*A2
            UA22R = CSR*A3
<span class="comment">*</span><span class="comment">
</span>            VB21 = -D1*SNL*B1 + CSL*B2
            VB22R = CSL*B3
<span class="comment">*</span><span class="comment">
</span>            AUA21 = ABS( SNR )*ABS( A1 ) + ABS( CSR )*ABS1( A2 )
            AVB21 = ABS( SNL )*ABS( B1 ) + ABS( CSL )*ABS1( B2 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           zero (2,1) elements of U'*A and V'*B.
</span><span class="comment">*</span><span class="comment">
</span>            IF( ( ABS1( UA21 )+ABS( UA22R ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.249"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( CMPLX( VB22R ), VB21, CSQ, SNQ, R )
            ELSE IF( ( ABS1( VB21 )+ABS( VB22R ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.251"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( CMPLX( UA22R ), UA21, CSQ, SNQ, R )
            ELSE IF( AUA21 / ( ABS1( UA21 )+ABS( UA22R ) ).LE.AVB21 /
     $               ( ABS1( VB21 )+ABS( VB22R ) ) ) THEN
               CALL <a name="CLARTG.254"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( CMPLX( UA22R ), UA21, CSQ, SNQ, R )
            ELSE
               CALL <a name="CLARTG.256"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( CMPLX( VB22R ), VB21, CSQ, SNQ, R )
            END IF
<span class="comment">*</span><span class="comment">
</span>            CSU = CSR
            SNU = -CONJG( D1 )*SNR
            CSV = CSL
            SNV = -CONJG( D1 )*SNL
<span class="comment">*</span><span class="comment">
</span>         ELSE
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           Compute the (1,1) and (1,2) elements of U'*A and V'*B,
</span><span class="comment">*</span><span class="comment">           and (1,1) element of |U|'*|A| and |V|'*|B|.
</span><span class="comment">*</span><span class="comment">
</span>            UA11 = CSR*A1 + CONJG( D1 )*SNR*A2
            UA12 = CONJG( D1 )*SNR*A3
<span class="comment">*</span><span class="comment">
</span>            VB11 = CSL*B1 + CONJG( D1 )*SNL*B2
            VB12 = CONJG( D1 )*SNL*B3
<span class="comment">*</span><span class="comment">
</span>            AUA11 = ABS( CSR )*ABS( A1 ) + ABS( SNR )*ABS1( A2 )
            AVB11 = ABS( CSL )*ABS( B1 ) + ABS( SNL )*ABS1( B2 )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">           zero (1,1) elements of U'*A and V'*B, and then swap.
</span><span class="comment">*</span><span class="comment">
</span>            IF( ( ABS1( UA11 )+ABS1( UA12 ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.281"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( VB12, VB11, CSQ, SNQ, R )
            ELSE IF( ( ABS1( VB11 )+ABS1( VB12 ) ).EQ.ZERO ) THEN
               CALL <a name="CLARTG.283"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( UA12, UA11, CSQ, SNQ, R )
            ELSE IF( AUA11 / ( ABS1( UA11 )+ABS1( UA12 ) ).LE.AVB11 /
     $               ( ABS1( VB11 )+ABS1( VB12 ) ) ) THEN
               CALL <a name="CLARTG.286"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( UA12, UA11, CSQ, SNQ, R )
            ELSE
               CALL <a name="CLARTG.288"></a><a href="clartg.f.html#CLARTG.1">CLARTG</a>( VB12, VB11, CSQ, SNQ, R )
            END IF
<span class="comment">*</span><span class="comment">
</span>            CSU = SNR
            SNU = CONJG( D1 )*CSR
            CSV = SNL
            SNV = CONJG( D1 )*CSL
<span class="comment">*</span><span class="comment">
</span>         END IF
<span class="comment">*</span><span class="comment">
</span>      END IF
<span class="comment">*</span><span class="comment">
</span>      RETURN
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     End of <a name="CLAGS2.302"></a><a href="clags2.f.html#CLAGS2.1">CLAGS2</a>
</span><span class="comment">*</span><span class="comment">
</span>      END

</pre>

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